Civil Engineering Reference
In-Depth Information
2
dy
dx
P
EI
=−
cr
y
2
P
EI
cr
y
′′ +
y
=
0
This is a second-order, linear, ordinary differential equation with the fol-
lowing general solution.
=
()
+
()
yA kx B x
cos
sin
P
EI
where
k
=
cr
The constants
A
and
B
can be evaluated using the boundary condition. At
x=
0
, y=
0, the equation becomes:
=
()
+
()
0
Ak Bk
cos
0
sin
0
This equation yields
A=
0 and at
x=L
,
y=
0, the equation becomes:
0 =
()
B L
sin
This condition is true when
B
sin(
kL
)
=
0, which can only be true for three
conditions as follows:
B =
0
No deflection
kL =
0
No load
kL = p,
2
p,
3
p, …, = np
where
n
= 1, 2, 3, …
Therefore, the following can be found and is the critical buckling load:
P
EI
Ln
kL
=
cr
=
p
P
EI
Ln
P
n I
L
cr
2
=
2
p
2
22
p
=
cr
2
